This is a handout from a first year graduate course in Complex Analysis. After reviewing the chain rule it moves on to the theory of differentiation for mappings between Euclidean spaces. In class (but not in these notes) I emphasized how the Cauchy-Riemann equations simply assert that the derivative of a map R^2->R^2 is a complex linear transformation. The notes go on to discuss conformality of the stereographic projection, and finally, via the complex logarithm, to relate this to the famous Mercator map projection.